## Element of Surface Area

In order to find an element of area
$d \mathbf{S}$
surface
$S$
, it is necessary to find vectors
$\vec{u}. \; \vec{v}$
in the surface then we can define an element of surface area as
$d \mathbf{S}= d \vec{u} \times d \vec{v}$
.
Because
$d \mathbf{S}$
is defined as the cross product of two vectors, it is itself a vector and is perpendicular to both
$d vec{u}$
and
$d \vec{v}$
.
Suppose the equation of a surface is
$z=a^2-x^2-y^2$
. Write this as
$a^2=x^2+y^2+z$
.
Then
$0=2xdx+2ydy+dz$
. Along the line where
$y=0$
this becomes
$0=2xdx+dz$
corresponding to the vector
$\begin{pmatrix}dx\\0\\-2xdx\end{pmatrix} = \begin{pmatrix}1\\0\\-2x\end{pmatrix} dx$
.
Along the line where
$x=0$
this becomes
$0=2ydy+dz$
corresponding to the vector
$\begin{pmatrix}0\\dy\\-2ydy\end{pmatrix} = \begin{pmatrix}0\\1\\-2y\end{pmatrix} dy$
.
The cross product of these two vectors is
$\begin{pmatrix}1\\0\\-2x\end{pmatrix} dx \times \begin{pmatrix}0\\1\\-2y\end{pmatrix} dy = \begin{pmatrix}2x\\2y\\1\end{pmatrix} dxdy$
.
This element of area is a vector perpendicular to the surface at the point
$(x,y,a^2-x^2-y^2$
.